monomvn (version 1.9-7)

# blasso: Bayesian Lasso/NG, Horseshoe, and Ridge Regression

## Description

Inference for ordinary least squares, lasso/NG, horseshoe and ridge regression models by (Gibbs) sampling from the Bayesian posterior distribution, augmented with Reversible Jump for model selection

## Usage

bhs(X, y, T=1000, thin=NULL, RJ=TRUE, M=NULL, beta=NULL,
lambda2=1, s2=var(y-mean(y)), mprior=0, ab=NULL,
theta=0, rao.s2=TRUE, icept=TRUE, normalize=TRUE, verb=1)
bridge(X, y, T = 1000, thin = NULL, RJ = TRUE, M = NULL,
beta = NULL, lambda2 = 1, s2 = var(y-mean(y)), mprior = 0,
rd = NULL, ab = NULL, theta=0, rao.s2 = TRUE, icept = TRUE,
normalize = TRUE, verb = 1)
blasso(X, y, T = 1000, thin = NULL, RJ = TRUE, M = NULL,
beta = NULL, lambda2 = 1, s2 = var(y-mean(y)),
case = c("default", "ridge", "hs", "ng"), mprior = 0, rd = NULL,
ab = NULL, theta = 0, rao.s2 = TRUE, icept = TRUE,
normalize = TRUE, verb = 1)

## Arguments

X

data.frame, matrix, or vector of inputs X

y

vector of output responses y of length equal to the leading dimension (rows) of X, i.e., length(y) == nrow(X)

T

total number of MCMC samples to be collected

thin

number of MCMC samples to skip before a sample is collected (via thinning). If NULL (default), then thin is determined based on the regression model implied by RJ, lambda2, and ncol(X); and also on the errors model implied by theta and nrow(X)

RJ

if TRUE then model selection on the columns of the design matrix (and thus the parameter beta in the model) is performed by Reversible Jump (RJ) MCMC. The initial model is specified by the beta input, described below, and the maximal number of covariates in the model is specified by M

M

the maximal number of allowed covariates (columns of X) in the model. If input lambda2 > 0 then any M <= ncol(X) is allowed. Otherwise it must be that M <= min(ncol(X), length(y)-1), which is default value when a NULL argument is given

beta

initial setting of the regression coefficients. Any zero-components will imply that the corresponding covariate (column of X) is not in the initial model. When input RJ = FALSE (no RJ) and lambda2 > 0 (use lasso) then no components are allowed to be exactly zero. The default setting is therefore contextual; see below for details

lambda2

square of the initial lasso penalty parameter. If zero, then least squares regressions are used

s2

initial variance parameter

case

specifies if ridge regression, the Normal-Gamma, or the horseshoe prior should be done instead of the lasso; only meaningful when lambda2 > 0

mprior

prior on the number of non-zero regression coefficients (and therefore covariates) m in the model. The default (mprior = 0) encodes the uniform prior on 0 <= m <= M. A scalar value 0 < mprior < 1 implies a Binomial prior Bin(m|n=M,p=mprior). A 2-vector mprior=c(g,h) of positive values g and h represents gives Bin(m|n=M,p) prior where p~Beta(g,h)

rd

=c(r, delta), the alpha (shape) parameter and $$\beta$$ (rate) parameter to the gamma distribution prior G(r,delta) for the $$\lambda^2$$ parameter under the lasso model; or, the $$\alpha$$ (shape) parameter and $$\beta$$ (scale) parameter to the inverse-gamma distribution IG(r/2, delta/2) prior for the $$\lambda^2$$ parameter under the ridge regression model. A default of NULL generates appropriate non-informative values depending on the nature of the regression. Specifying rd=FALSE causes lambda2 values to be fixed at their starting value, i.e., not sampled. See the details below for information on the special settings for ridge regression

ab

=c(a, b), the $$\alpha$$ (shape) parameter and the $$\beta$$ (scale) parameter for the inverse-gamma distribution prior IG(a,b) for the variance parameter s2. A default of NULL generates appropriate non-informative values depending on the nature of the regression

theta

the rate parameter (> 0) to the exponential prior on the degrees of freedom paramter nu under a model with Student-t errors implemented by a scale-mixture prior. The default setting of theta = 0 turns off this prior, defaulting to a normal errors prior

rao.s2

indicates whether Rao-Blackwellized samples for $$\sigma^2$$ should be used (default TRUE); see below for more details

icept

if TRUE, an implicit intercept term is fit in the model, otherwise the the intercept is zero; default is TRUE

normalize

if TRUE, each variable is standardized to have unit L2-norm, otherwise it is left alone; default is TRUE

verb

verbosity level; currently only verb = 0 and verb = 1 are supported

## Value

blasso returns an object of class "blasso", which is a list containing a copy of all of the input arguments as well as of the components listed below.

call

a copy of the function call as used

mu

a vector of T samples of the (un-penalized) “intercept” parameter

beta

a T*ncol(X) matrix of T samples from the (penalized) regression coefficients

m

the number of non-zero entries in each vector of T samples of beta

s2

a vector of T samples of the variance parameter

lambda2

a vector of T samples of the penalty parameter

gamma

a vector of T with the gamma parameter when case = "ng"

tau2i

a T*ncol(X) matrix of T samples from the (latent) inverse diagonal of the prior covariance matrix for beta, obtained for Lasso regressions

omega2

a T*nrow(X) matrix of T samples from the (latent) diagonal of the covariance matrix of the response providing a scale-mixture implementation of Student-t errors with degrees of freedom nu when active (input theta > 0)

nu

a vector of T samples of the degrees of freedom parameter to the Student-t errors mode when active (input theta > 0)

pi

a vector of T samples of the Binomial proportion p that was given a Beta prior, as described above for the 2-vector version of the mprior input

lpost

the log posterior probability of each (saved) sample of the joint parameters

llik

the log likelihood of each (saved) sample of the parameters

llik.norm

the log likelihood of each (saved) sample of the parameters under the Normal errors model when sampling under the Student-t model; i.e., it is not present unless theta > 0

## Details

The Bayesian lasso model and Gibbs Sampling algorithm is described in detail in Park & Casella (2008). The algorithm implemented by this function is identical to that described therein, with the exception of an added “option” to use a Rao-Blackwellized sample of $$\sigma^2$$ (with $$\beta$$ integrated out) for improved mixing, and the model selections by RJ described below. When input argument lambda2 = 0 is supplied, the model is a simple hierarchical linear model where $$(\beta,\sigma^2)$$ is given a Jeffrey's prior

Specifying RJ = TRUE causes Bayesian model selection and averaging to commence for choosing which of the columns of the design matrix X (and thus parameters beta) should be included in the model. The zero-components of the beta input specify which columns are in the initial model, and M specifies the maximal number of columns.

The RJ mechanism implemented here for the Bayesian lasso model selection differs from the one described by Hans (2009), which is based on an idea from Geweke (1996). Those methods require departing from the Park & Casella (2008) latent-variable model and requires sampling from each conditional $$\beta_i | \beta_{(-i)}, \dots$$ for all $$i$$, since a mixture prior with a point-mass at zero is placed on each $$\beta_i$$. Out implementation here requires no such special prior and retains the joint sampling from the full $$\beta$$ vector of non-zero entries, which we believe yields better mixing in the Markov chain. RJ proposals to increase/decrease the number of non-zero entries does proceed component-wise, but the acceptance rates are high due due to marginalized between-model moves (Troughton & Godsill, 1997).

When the lasso prior or RJ is used, the automatic thinning level (unless thin != NULL) is determined by the number of columns of X since this many latent variables are introduced

Bayesian ridge regression is implemented as a special case via the bridge function. This essentially calls blasso with case = "ridge". A default setting of rd = c(0,0) is implied by rd = NULL, giving the Jeffery's prior for the penalty parameter $$\lambda^2$$ unless ncol(X) >= length(y) in which case the proper specification of rd = c(5,10) is used instead.

The Normal--Gamma prior (Griffin & Brown, 2009) is implemented as an extension to the Bayesian lasso with case = "ng". Many thanks to James Scott for providing the code needed to extend the method(s) to use the horseshoe prior (Carvalho, Polson, Scott, 2010).

When theta > 0 then the Student-t errors via scale mixtures (and thereby extra latent variables omega2) of Geweke (1993) is applied as an extension to the Bayesian lasso/ridge model. If Student-t errors are used the automatic thinning level is augmented (unless thin != NULL) by the number of rows in X since this many latent variables are introduced

## References

Park, T., Casella, G. (2008). The Bayesian Lasso. Journal of the American Statistical Association, 103(482), June 2008, pp. 681-686 http://www.stat.ufl.edu/~casella/Papers/Lasso.pdf

Griffin, J.E. and Brown, P.J. (2009). Inference with Normal-Gamma prior distributions in regression problems. Bayesian Analysis, 5, pp. 171-188. http://projecteuclid.org/euclid.ba/1340369797

Hans, C. (2009). Bayesian Lasso regression. Biometrika 96, pp. 835-845. http://biomet.oxfordjournals.org/content/96/4/835.abstract

Carvalho, C.M., Polson, N.G., and Scott, J.G. (2010) The horseshoe estimator for sparse signals. Biometrika 97(2): pp. 465-480. http://ftp.stat.duke.edu/WorkingPapers/08-31.pdf

Geweke, J. (1996). Variable selection and model comparison in regression. In Bayesian Statistics 5. Editors: J.M. Bernardo, J.O. Berger, A.P. Dawid and A.F.M. Smith, 609-620. Oxford Press.

Paul T. Troughton and Simon J. Godsill (1997). A reversible jump sampler for autoregressive time series, employing full conditionals to achieve efficient model space moves. Technical Report CUED/F-INFENG/TR.304, Cambridge University Engineering Department.

Geweke, J. (1993) Bayesian treatment of the independent Student-t linear model. Journal of Applied Econometrics, Vol. 8, S19-S40

http://bobby.gramacy.com/r_packages/monomvn

lm , lars in the lars package, regress, lm.ridge in the MASS package

## Examples

# NOT RUN {
## following the lars diabetes example
data(diabetes)
attach(diabetes)

## Ordinary Least Squares regression
reg.ols <- regress(x, y)

## Lasso regression
reg.las <- regress(x, y, method="lasso")

## Bayesian Lasso regression
reg.blas <- blasso(x, y)

## summarize the beta (regression coefficients) estimates
plot(reg.blas, burnin=200)
points(drop(reg.las$b), col=2, pch=20) points(drop(reg.ols$b), col=3, pch=18)
legend("topleft", c("blasso-map", "lasso", "lsr"),
col=c(2,2,3), pch=c(21,20,18))

## plot the size of different models visited
plot(reg.blas, burnin=200, which="m")

## get the summary
s <- summary(reg.blas, burnin=200)

## calculate the probability that each beta coef != zero
s$bn0 ## summarize s2 plot(reg.blas, burnin=200, which="s2") s$s2

## summarize lambda2
plot(reg.blas, burnin=200, which="lambda2")
s$lambda2 # } # NOT RUN { ## fit with Student-t errors ## (~400-times slower due to automatic thinning level) regt.blas <- blasso(x, y, theta=0.1) ## plotting some information about nu, and quantiles plot(regt.blas, "nu", burnin=200) quantile(regt.blas$nu[-(1:200)], c(0.05, 0.95))

## Bayes Factor shows strong evidence for Student-t model
mean(exp(regt.blas$llik[-(1:200)] - regt.blas$llik.norm[-(1:200)]))
# }
# NOT RUN {
## clean up
detach(diabetes)
# }